Polarization and Gorenstein liaison

TL;DR

This paper explores Gorenstein liaison theory, focusing on constructing G-links via polarization techniques, and demonstrates that polarizations of stable Cohen--Macaulay monomial ideals have vertex decomposable Stanley--Reisner complexes.

Contribution

It introduces methods to lift G-links from reduced schemes to more general schemes using polarization and vertex decomposition, linking combinatorial and algebraic properties.

Findings

Polarizations of stable Cohen--Macaulay monomial ideals have vertex decomposable complexes.

Conditions are established for lifting G-links via polarization.

A relationship between polarization of Gr"obner bases and G-biliaison is demonstrated.

Abstract

A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.